I cannot find a proof of this theorem. May anyone assist?
$p_{n+1}-p_n\gg\frac{\log \log \log p_n}{\log \log \log \log p_n} \log{p_n}$
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1$\begingroup$ I'm voting to close this question because one cannot prove a false assertion. $\endgroup$– Stefan Kohl ♦Commented Sep 11, 2015 at 9:08
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1$\begingroup$ Probably OP means that this happens infinitely often. $\endgroup$– Fedor PetrovCommented Sep 11, 2015 at 12:49
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$\begingroup$ I am voting to close as Googling "long gaps between primes" yields the state of the art. $\endgroup$– Boris BukhCommented Sep 11, 2015 at 13:09
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$\begingroup$ The lower bound of Westzynthius involves sifting an interval $[R, R+ p_n\xi]$ with the first n primes in three stages: cross out multiples of all of the (k+1)st through lth primes, then choose residues to maximally sieve the remaining with the first k primes. This leaves much fewer than n-l holes in the interval to be covered by the remaining primes. $\xi$ in the paper is "like" a constant times $\frac{\log \log p_n}{\log \log \log p_n}$. I intend to post a review of the lower bound argument eventually. Gerhard "Still Playing With Upper Bound" Paseman, 2015.09.11 $\endgroup$– Gerhard PasemanCommented Sep 11, 2015 at 21:07
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1 Answer
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The bound as stated is false, because not all prime gaps are that large. In fact we know since the work of Yitang Zhang (2013) that there are infinitely many bounded prime gaps.
The state of the art regarding (occasional) large prime gaps is contained in the work of Ford-Green-Konyagin-Maynard-Tao. Consulting the references in this paper should lead you to earlier but still rather strong results, e.g. the work of Rankin.
answered Sep 10, 2015 at 23:28
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2$\begingroup$ The bound is reminiscent of Westzynthius (1931). A reference for this work can be found at mathoverflow.net/questions/37679 . Gerhard "And A Few Other Places" Paseman, 2015.09.10 $\endgroup$ Commented Sep 10, 2015 at 23:56